for problem one i think my idea of usin bolzano-weistrass theorem is not so bad because it is probable that most of the theorem used to proove it other ways use need this theorem (may be even the cauchy sequence) and also because whathever strange and arbitrary you would had cut you square in pieces select one pieces and then cut it in up again ect...this theorem prooves that you got a convergent serie (it would be nice to try to proove the arbitrary cuting problem whithout using the bolzano_weistrass théorème anyway)

for problem two i think it is useless to proove anything about U S and T when they are the empty set because they have no element so we cannot say anything about the product of two element of such set.
usualy set closed by order are groups (which are none empty) or that sort of thing
beter look to the definition of a set closedby multiplication to see if it admits empty set by definition but demonstration is useless about what do or do not elements of the empy set!

for number 3 i found only one trivial (x=1) solution of the hypotesisis of recurence for n=1
it would be nice to have an other exemple of x to have a concrete idea of what strong induction is